Let \$Lambdaleft(n ight)\$ be the Von Mangoldt function, let [r_{G}left(n ight):=\underset{{scriptstyle m_{1}+m_{2}=n}}{sum_{m_{1},m_{2}leq n}}Lambdaleft(m_{1} ight)Lambdaleft(m_{2} ight),] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let \$N>2\$ be an integer. We will find the explicit formulae for the average of \$r_{G}left(n ight)\$ in terms of elementary functions, the incomplete Beta function \$B_{z}left(a,b ight)\$,series over \$ ho\$ that, with or without subscript, runs over the non-trivial zeros of the Riemann Zeta function and the Dilogarithm function. We will also prove the explicit formulae in an asymptotic form and a truncated formula for the average of \$r_{G}left(n ight)\$. Some observation about these formulae and the average with Cesàro weight [ rac{1}{Gammaleft(k+1 ight)}sum_{nleq N}r_{G}left(n ight)left(N-n ight)^{k},,k>0] and [r_{PT}left(N,h ight):=sum_{n=0}^{N}Lambdaleft(n ight)Lambdaleft(n+h ight),,hinmathbb{N}] are included.

### Explicit formula for the average of goldbach numbers

#### Abstract

Let \$Lambdaleft(n ight)\$ be the Von Mangoldt function, let [r_{G}left(n ight):=\underset{{scriptstyle m_{1}+m_{2}=n}}{sum_{m_{1},m_{2}leq n}}Lambdaleft(m_{1} ight)Lambdaleft(m_{2} ight),] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let \$N>2\$ be an integer. We will find the explicit formulae for the average of \$r_{G}left(n ight)\$ in terms of elementary functions, the incomplete Beta function \$B_{z}left(a,b ight)\$,series over \$ ho\$ that, with or without subscript, runs over the non-trivial zeros of the Riemann Zeta function and the Dilogarithm function. We will also prove the explicit formulae in an asymptotic form and a truncated formula for the average of \$r_{G}left(n ight)\$. Some observation about these formulae and the average with Cesàro weight [ rac{1}{Gammaleft(k+1 ight)}sum_{nleq N}r_{G}left(n ight)left(N-n ight)^{k},,k>0] and [r_{PT}left(N,h ight):=sum_{n=0}^{N}Lambdaleft(n ight)Lambdaleft(n+h ight),,hinmathbb{N}] are included.
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2019
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11391/1455286`
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